Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $p \neq 0$. $z = \dfrac{p + 5}{p^2 + 9p + 20} \div \dfrac{p + 1}{p + 4} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $z = \dfrac{p + 5}{p^2 + 9p + 20} \times \dfrac{p + 4}{p + 1} $ First factor the quadratic. $z = \dfrac{p + 5}{(p + 4)(p + 5)} \times \dfrac{p + 4}{p + 1} $ Then multiply the two numerators and multiply the two denominators. $z = \dfrac{ (p + 5) \times (p + 4) } { (p + 4)(p + 5) \times (p + 1) } $ $z = \dfrac{ (p + 5)(p + 4)}{ (p + 4)(p + 5)(p + 1)} $ Notice that $(p + 5)$ and $(p + 4)$ appear in both the numerator and denominator so we can cancel them. $z = \dfrac{ (p + 5)\cancel{(p + 4)}}{ \cancel{(p + 4)}(p + 5)(p + 1)} $ We are dividing by $p + 4$ , so $p + 4 \neq 0$ Therefore, $p \neq -4$ $z = \dfrac{ \cancel{(p + 5)}\cancel{(p + 4)}}{ \cancel{(p + 4)}\cancel{(p + 5)}(p + 1)} $ We are dividing by $p + 5$ , so $p + 5 \neq 0$ Therefore, $p \neq -5$ $z = \dfrac{1}{p + 1} ; \space p \neq -4 ; \space p \neq -5 $